Semipositivity with respect to the Lorentz cone

Abstract

Lorentz cone in the Euclidean space $\mathbb{R}^{n}$ is defined as $\mathcal{L}^n_+:=\{x\in \mathbb{R}^n:x_n\geq 0,\sum\limits_{i=1}^{n-1}x_{i}^2\leq x_n^2\}$. The paper aims to study semipositivity of matrices with respect to $\mathcal{L}^n_+$. A $n\times n$ real matrix $A$ is $\mathcal{L}^n_+$-semipositive if there exists $x\in \mathcal{L}^n_+$ such that $Ax\in \mathop{\rm int} (\mathcal{L}^n_+)$ (the topological interior of $\mathcal{L}^n_+$). $\mathcal{L}^n_+$-positive matrices ($A(\mathcal{L}^n_+\setminus\{0\})\subseteq \mathcal{L}^n_+$) and minimally $\mathcal{L}^n_+$-semipositive matrices ($A^{-1}(\mathcal{L}^n_+)\subseteq \mathcal{L}^n_+$) are two important subclasses of $\mathcal{L}^n_+$-semipositive matrices. In this paper, we establish the existence of bases for the real vector space of $n\times n$ matrices, consisting of $\mathcal{L}^n_+$-positive matrices and of minimally $\mathcal{L}^n_+$-semipositive matrices. Sufficient conditions are determined for $\mathcal{L}^{n}_+$-semipositivity, in terms of the length of rows(columns) of the matrices. Furthermore, we discuss properties of $\mathcal{L}^n_+$-semipositive matrices involving product of matrices. At last, $\mathcal{L}^2_+$-semipositive matrices are described via entries of the matrices and equivalent $\mathcal{L}^n_+$-semipositive matrices are studied.